A277791 Denominator of sum of reciprocals of proper divisors of n.
1, 1, 1, 2, 1, 6, 1, 4, 3, 10, 1, 4, 1, 14, 15, 8, 1, 9, 1, 20, 21, 22, 1, 24, 5, 26, 9, 28, 1, 30, 1, 16, 33, 34, 35, 2, 1, 38, 39, 40, 1, 42, 1, 44, 45, 46, 1, 16, 7, 25, 51, 52, 1, 54, 55, 8, 57, 58, 1, 60, 1, 62, 63, 32, 65, 6, 1, 68, 69, 70, 1, 36, 1, 74, 25, 76, 77, 78, 1, 16
Offset: 1
Examples
a(4) = 2 because 4 has 3 divisors {1,2,4} therefore 2 proper divisors {1,2} and 1/1 + 1/2 = 3/2.
0, 1, 1, 3/2, 1, 11/6, 1, 7/4, 4/3, 17/10, 1, 9/4, 1, 23/14, 23/15, 15/8, 1, 19/9, 1, 41/20, 31/21, 35/22, 1, 59/24, 6/5, 41/26, 13/9, 55/28, ...
Links
- Robert Israel, Table of n, a(n) for n = 1..10000
- Eric Weisstein's World of Mathematics, Restricted Divisor Function
- Index entries for sequences related to sums of divisors
Crossrefs
Programs
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Maple
with(numtheory): P:=proc(n) local a,k; a:=divisors(n) minus {n}; denom(add(1/a[k],k=1..nops(a))); end: seq(P(i),i=1..80); # Paolo P. Lava, Oct 17 2018 -
Mathematica
Table[Denominator[DivisorSigma[-1, n] - 1/n], {n, 1, 80}] Table[Denominator[(DivisorSigma[1, n] - 1)/n], {n, 1, 80}] -
PARI
a(n) = denominator((sigma(n)-1)/n); \\ Michel Marcus, Nov 01 2016
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Python
from math import gcd from sympy import divisor_sigma def A277791(n): return n//gcd(n,divisor_sigma(n)-1) # Chai Wah Wu, Jul 18 2022
Formula
a(n) = denominator(Sum_{d|n, d
a(n) = denominator((sigma_1(n)-1)/n).
a(p) = 1 when p is prime.
a(p^k) = p^(k-1).
Dirichlet g.f.: (zeta(s) - 1)*zeta(s+1) (for fraction Sum_{d|n, d
A281085 Numerator of sum of reciprocals of numbers less than n that do not divide n.
0, 0, 1, 1, 13, 9, 29, 59, 1163, 569, 4861, 21341, 58301, 79139, 619181, 260041, 1715839, 1808487, 10190221, 116220883, 32925391, 966183, 13920029, 455451475, 4597423223, 1536962359, 64517796001, 154777722503, 235091155703, 3714867879427, 6975593267347, 75441657715841
Offset: 1
Examples
a(6) = 9 because 6 has 4 divisors {1,2,3,6} therefore 2 non-divisors {4,5} and 1/4 + 1/5 = 9/20.
0, 0, 1/2, 1/3, 13/12, 9/20, 29/20, 59/70, 1163/840, 569/504, 4861/2520, 21341/27720, 58301/27720, 79139/51480, 619181/360360, 260041/180180, ...
Crossrefs
Programs
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Mathematica
Table[Numerator[HarmonicNumber[n] - DivisorSigma[-1, n]], {n, 1, 32}] Table[Numerator[HarmonicNumber[n] - DivisorSigma[1, n]/n], {n, 1, 32}]
A372836 a(n) is the numerator of Sum_{d|n, 1 < d < n} 1/d.
0, 0, 0, 1, 0, 5, 0, 3, 1, 7, 0, 5, 0, 9, 8, 7, 0, 10, 0, 21, 10, 13, 0, 35, 1, 15, 4, 27, 0, 41, 0, 15, 14, 19, 12, 3, 0, 21, 16, 49, 0, 53, 0, 39, 32, 25, 0, 25, 1, 21, 20, 45, 0, 65, 16, 9, 22, 31, 0, 107, 0, 33, 40, 31, 18, 7, 0, 57, 26, 73, 0, 61, 0, 39, 16, 63, 18, 89, 0, 21
Offset: 1
Examples
0, 0, 0, 1/2, 0, 5/6, 0, 3/4, 1/3, 7/10, 0, 5/4, 0, 9/14, 8/15, 7/8, 0, 10/9, ...
Links
- Antti Karttunen, Table of n, a(n) for n = 1..16384
Programs
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Mathematica
nmax = 80; CoefficientList[Series[Sum[x^(2 k)/(k (1 - x^k)), {k, 2, nmax}], {x, 0, nmax}], x] // Numerator // Rest -
PARI
a(n) = numerator(sumdiv(n, d, if ((d>1) && (d
Formula
Numerators of coefficients in expansion of Sum_{k>=2} x^(2*k) / (k * (1 - x^k)).